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        <h1 id="排序"><a href="#排序" class="headerlink" title="排序"></a>排序</h1><h2 id="知识图谱"><a href="#知识图谱" class="headerlink" title="知识图谱"></a>知识图谱</h2><p><img src="%2_D3I`7JCA{0@V421[B2Y6.png" alt="img"></p>
<h2 id="基本概念"><a href="#基本概念" class="headerlink" title="基本概念"></a>基本概念</h2><h3 id="排序定义"><a href="#排序定义" class="headerlink" title="排序定义"></a>排序定义</h3><h3 id="稳定性"><a href="#稳定性" class="headerlink" title="稳定性"></a>稳定性</h3><h3 id="衡量标准-时、空复杂度"><a href="#衡量标准-时、空复杂度" class="headerlink" title="衡量标准:时、空复杂度"></a>衡量标准:时、空复杂度</h3><h2 id="内部排序"><a href="#内部排序" class="headerlink" title="内部排序"></a>内部排序</h2><h3 id="插入徘序"><a href="#插入徘序" class="headerlink" title="插入徘序"></a>插入徘序</h3><h4 id="直接插入排序"><a href="#直接插入排序" class="headerlink" title="直接插入排序"></a>直接插入排序</h4><p>查找出L(i)在L[1……i-1]中的插入位置k<br>将L[k……i-1]中的所有元素全部后移一个位置<br>将L(i)复制到L(k)</p>
<h4 id="折半插入排序"><a href="#折半插入排序" class="headerlink" title="折半插入排序"></a>折半插入排序</h4><p>从上述算法中，不难看出折半括入排仅减小了比较元素的次数，约为O(nlog2n)，该比较次数与待排序表的初始状态无关，仅取决干表中的元素个数n;而元素的移动次数并未改变，它依赖于待排序表的初始状态。因此,折半插入排序的时间复杂度仍为O(n2)。折半插入排序是一种稳定的排序方法。<br><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">InsertSort</span><span class="params">(ElemType A[],<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function">    <span class="keyword">int</span> i, j,low,high, mid</span>;</span><br><span class="line">    <span class="keyword">for</span>(i=<span class="number">2</span>;i&lt;=n;i++)&#123;</span><br><span class="line">    <span class="comment">//依次将A[2]~A[n]插入前面的已排序序列</span></span><br><span class="line">        A[<span class="number">0</span>]=A[i];</span><br><span class="line">        <span class="comment">//将A[i]暂存到A[0]</span></span><br><span class="line">        low=<span class="number">1</span>;high=i<span class="number">-1</span>;</span><br><span class="line">        <span class="comment">//设置折半查找的范围</span></span><br><span class="line">        <span class="keyword">while</span>(low&lt;=high)&#123;</span><br><span class="line">            <span class="comment">//折半查找(默认递增有序)</span></span><br><span class="line">            mid=(low+high)/<span class="number">2</span>;</span><br><span class="line">            <span class="comment">//取中间点</span></span><br><span class="line">            <span class="keyword">if</span>(A[mid].key&gt;A[<span class="number">0</span>].key)high=mid<span class="number">-1</span>;</span><br><span class="line">            <span class="comment">//查找左半子表</span></span><br><span class="line">            <span class="keyword">else</span> low=mid+<span class="number">1</span></span><br><span class="line">            <span class="comment">//查找右半子表</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">for</span>(j=i<span class="number">-1</span>;j&gt;=high+l;--j)</span><br><span class="line">            A[j+<span class="number">1</span>J=A[j];</span><br><span class="line">        <span class="comment">//统一后移元素，空出插入位置</span></span><br><span class="line">        A[high+<span class="number">1</span>]=A[<span class="number">0</span>];</span><br><span class="line">        <span class="comment">//插入操作</span></span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></p>
<h4 id="希尔排序"><a href="#希尔排序" class="headerlink" title="希尔排序"></a>希尔排序</h4><p>希尔排序的基本思想是:先将待排序表分割成若干形如 L[i, i+ d, i + 2d,…, i + kd]的“特殊”子表，分别进行直接插入排序，当整个表中的元素已呈“基本有序”时，再对全体记录进行一次直接插入排序。<br>希尔排序的排序过程如下:先取一个小于n的步长d,把表中的全部记录分成d组，所有距离为d的倍数的记录放在同一组中，在各组中进行直接插入排序;然后取第二个步长d,<d,重复上述过程，直到所取到的d, = 1，即所有记录已放在同一组中，再进行直接插入排序，由于此时已经具有较好的局部有序性，故可以很快得到最终结果。到目前为止，尚未求得一个最好的增量序列，希尔提出的方法是d =n/2，d+1=Ld/2],并且最后一个增量等于1.
空间效率：O(1)
时间效率：O(n^1.3) O(n^2)
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">shellSort</span> <span class="params">(ElemType A[],<span class="keyword">int</span> n)</span></span>&#123;</span><br><span class="line"><span class="comment">//对顺序表做希尔抽八排了，伞早法N且按插入排序相比，做了以下修改:</span></span><br><span class="line"><span class="comment">//1.前后记录位置的增量是dk，不是1</span></span><br><span class="line"><span class="comment">//2.A[0]只是暂存单元，个是哨兵，当j&lt;=0时，插入位置已到</span></span><br><span class="line">    <span class="comment">//步长变化</span></span><br><span class="line">    <span class="keyword">for</span>(dk=n/<span class="number">2</span>;dk&gt;=l;dk=dk/<span class="number">2</span>)</span><br><span class="line">        <span class="keyword">for</span>(i=dk+<span class="number">1</span>;i&lt;=n;++i)</span><br><span class="line">            <span class="comment">//需将A[i]插入有序增量子表</span></span><br><span class="line">            <span class="keyword">if</span>(A[i].key&lt;A[i-dk].key)&#123;</span><br><span class="line">                <span class="comment">//暂存在A[0]</span></span><br><span class="line">                A[<span class="number">0</span>]=A[i];</span><br><span class="line">                <span class="comment">//记录后移,杳找插入的位置</span></span><br><span class="line">                <span class="keyword">for</span>(j=i-dk;j&gt;<span class="number">0</span>&amp;&amp;A[<span class="number">0</span>].key&lt;A[i].key;j-=dk)</span><br><span class="line">                    A[j+dk] =A[j];</span><br><span class="line">                <span class="comment">//插入</span></span><br><span class="line">                A[j+dk] =A[<span class="number">0</span>];</span><br><span class="line">            &#125;<span class="comment">//if</span></span><br></pre></td></tr></table></figure></p>
<h3 id="交换排序"><a href="#交换排序" class="headerlink" title="交换排序"></a>交换排序</h3><p> <img src="1.png" alt="1"><br><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Bubblesort</span><span class="params">(ElemType A[],<span class="keyword">int</span> n)</span></span>&#123;</span><br><span class="line"><span class="comment">//用冒泡排序法将序列A中的元素按从小到大排列</span></span><br><span class="line"><span class="keyword">for</span>(i=<span class="number">0</span>;i&lt;n-l;i++)&#123;</span><br><span class="line"><span class="comment">//表示本趟冒泡具否发生交换的标志</span></span><br><span class="line">flag=<span class="literal">false</span>;</span><br><span class="line"><span class="comment">//一趟冒泡过程</span></span><br><span class="line"><span class="keyword">for</span>(j=n<span class="number">-1</span>;j&gt;i;j--)</span><br><span class="line"><span class="keyword">if</span>(A[j<span class="number">-1</span>].key&gt;A[j].key)&#123;</span><br><span class="line"><span class="built_in">swap</span>(A[j<span class="number">-1</span>],A[<span class="number">1</span>]);/交换</span><br><span class="line">flag=<span class="literal">true</span>;</span><br><span class="line">&#125;</span><br><span class="line"><span class="keyword">if</span>(flag==<span class="literal">false</span>)</span><br><span class="line"><span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure><br>冒泡排序算法的性能分析如下:<br>空间效率:仅使用了常数个辅助单元，因而空间复杂度为O(1)。<br>时间效率:当初始序列有序时，显然第一趟冒泡后flag依然为false （本题冒泡没有元素交换),从而直接跳出循环,比较次数n-1,移动次数为0.从而最好情况下的时间复杂度为O(n);当初始序列为逆序时，需要进行n-1趟排序，第i趟排序要进行n-i次关键字的比较，而且每次比较都必须移动元素3次来交换元素位置。这种情况下，</p>
<h4 id="冒泡排序"><a href="#冒泡排序" class="headerlink" title="冒泡排序"></a>冒泡排序</h4><h4 id="快速排序"><a href="#快速排序" class="headerlink" title="快速排序"></a>快速排序</h4><h3 id="选择排序"><a href="#选择排序" class="headerlink" title="选择排序"></a>选择排序</h3><h4 id="简单选择排序"><a href="#简单选择排序" class="headerlink" title="简单选择排序"></a>简单选择排序</h4><h4 id="堆排序"><a href="#堆排序" class="headerlink" title="堆排序"></a>堆排序</h4><h3 id="归并排序"><a href="#归并排序" class="headerlink" title="归并排序"></a>归并排序</h3><h3 id="基数排序"><a href="#基数排序" class="headerlink" title="基数排序"></a>基数排序</h3><h2 id="外部排序一—多路归并排序"><a href="#外部排序一—多路归并排序" class="headerlink" title="外部排序一—多路归并排序"></a>外部排序一—多路归并排序</h2>
      
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